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15   WEAK TOPOLOGIES       77

(12.15.9)   Let E be a separable normed space. Then every closed ball B':
\\x'\\ ^ r in E' is metrizable and compact with respect to the weak topology.

By (12.15.7) and ( it is enough to show that B' is weakly closed,
and this follows from the fact that the function JC'H||JC'|| is lower semi-
continuous on E' (12.7.2).

Now let E be a Hilbert space (6.2). For each xe E, let/*) denote the
continuous linear form y*-+(y\x) on E. It follows from (6.3.2) that x\-*j(x) is
a semilinear isometry (i.e., we have

for all scalars X) of E onto its dual E'. We can therefore "transport" to E the
weak topology on E' : the weak topology on E is therefore the topology defined
by the seminorms x\~+ \(a \ x)\ as a runs through E. This topology is coarser
than the topology defined by the norm on E (which is called the strong
topology of the Hilbert space E). From (12.15.9) we have:

(12.15.10) In a separable Hilbert space, every closed ball is metrizable and
compact with respect to the weak topology.

If w is a continuous endomorphism (with respect to the strong topology) of
a Hilbert space E, then by virtue of (12.15.4) and (11.5.1)

(u(x)\y) = <(
for all x,y in E, and therefore (6.3.2)

(12.15.11)                              w*=7~1 o'Wo/

This relation shows immediately that '# is strongly continuous and that
H't/ll = ||w*|| = \\u\\. Replacing u by u* in (12.15.11), we see that every strongly
continuous endomorphism of E is also weakly continuous (cf, (12.16.7)).

(1 2.1 5.1 2)   In a Hilbert space E, a sequence (xn) converges strongly to a point a
if and only if it converges weakly to a and Urn \\xn\\ = \\a\\.

n~+ oo

The conditions are clearly necessary. To see that they are sufficient, con-
sider the formula

Since by hypothesis the sequence ((xn \ a)) converges to (a \ a) = || a\\2, it follows
that llx. - all tends to 0.eak topology on E'.